Volterra operator

Volterra operator

In mathematics, in the area of functional analysis and operator theory, the Volterra operator represents the operation of indefinite integration, viewed as a bounded linear operator on the space "L"2(0,1) of complex-valued square integrable functions on the interval (0,1). It is the operator corresponding to the Volterra integral equations.

Definition

The Volterra operator "V" may be defined at a function x(s) in L^2 left(0, 1 ight) and a value t in left(0, 1 ight)

:V(x)(t) = int_0^t{x(s), ds}.

Properties

*"V" is a bounded linear operator between Hilbert spaces, with Hermitian adjoint

::V^*(x)(t) = int_t^1{x(s), ds}.
*"V" is a Hilbert-Schmidt operator, hence in particular is compact.
*"V" has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ("V") = {0}.
*"V" is a quasinilpotent operator (that is, the spectral radius, "ρ"("V"), is zero), but it is not nilpotent.
*The operator norm of "V" is exactly ||"V"|| = 2⁄π.


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